Question

Factorize:

x³ + y³ + z³ - 3xyx

• Answer: ( 1/2 ) * ( x + y + z ) [ (x - y )² + ( y - z )² + ( z - x )² ]

Answer 1

x³ + y³ + z³ - 3xyz 
= (x + y + z)(x^2 + y^2 + z^2) - xy(x + y + z) - yz(x + y + z) - zx(x + y + z) 
= (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)

Answer 2

HELLO. ...

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We know,

x³ + y³ + z³ - 3xyz = (x + y + z){x² + y² + z² - xy - yz - zx }

=1/2 (x + y + z){2x² + 2y² + 2z² -2xy - 2yz - 2zx }

= 1/2 (x + y + z){( x² -2xy + y²) + (y² - 2yz + z²) +(z² - 2zx + x²) }

= 1/2 (x + y + z){(x - y)² + (y - z)² + (z - x)²}

Hence,varified///

Now,

64x³ + 125y³ - 64z³ +240xyz

= (4x)³ + (5y)³ +(-4z)³ -3(4x)(5y)(-4z)

This is just like above Identity , use that .

=1/2 (4x + 5y -4z ){(4x -5y)² + (5y+4z)² +(4x + 4z)² }

Hence, (4x + 5y -4z ) and {(4x -5y)² + (5y+4z)² +(4x + 4z)² } are the factors of given expression.